Numerical simulation of turbulent gas-particle flow in a riser using a quadrature-based moment method

Gas-particle flows are used in many industrial applications in the energy, oil and gas fields, such as coal gasification, production of light hydrocarbons by fluid catalytic cracking, catalytic combustion and different treatments aiming to reduce or eliminate pollutants. The particle phase of a gas-particle flow is described by analogy to a granular gas, by finding an approximate solution of the kinetic equation in the velocity-based number density function. In the recent past, many studies have been published on the mathematical modeling of gas-particle flows using hydrodynamic models (e.g. Enwald et al. 1996), where Navier-Stokes-type equations are solved to describe the particle phase as a continuum, computing its stress tensor using moment closures from kinetic theory (Gidaspow 1994). These closures, however, are obtained assuming that the flow is dominated by collisions and near equilibrium, which corresponds to considering a very small particle-phase Knudsen number. This assumption leads to inconsistencies and erroneous predictions of physical phenomena when these models are applied to dilute fluid-particle flows, where rarefaction effects are not negligible. In these flows, the wall Knudsen layers extend inside the bulk of the fluid, and cannot be accounted for with the simple addition of partial-slip boundary conditions. Recently Desjardin et al. (2008) showed that two-fluid models are unable to correctly capture particle trajectory crossing, seriously compromising their ability to correctly describe any velocity moment for finite Stokes numbers. These authors clarified that the particle segregation captured by two-fluid models for finite Knudsen numbers is artificially high due to their mathematical formulation, which leads to the formation of delta-shocks. In order to overcome these shortcomings, Fox (2008) developed a third-order quadrature-based moment method for dilute gas-particle flows, which has been successfully coupled to a fluid solver to compute dilute and moderately dilute gas-particle flows by Passalacqua et al. (2010) in two dimensions. These authors validated their model against Euler-Lagrange and two-fluid simulations. In this work, the fully coupled quadrature-based fluid-particle code described in Passalacqua et al. (2010) is applied to simulate turbulent gas-particle flow in the riser described by He et al. (2009), using a three-dimensional configuration. This application shows the predictive capabilities and the robustness of the quadrature-based moment method to predict the behavior of gas-particle flows in accordance with experiments (He et al. 2009)

Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier−Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of mono- and bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar−Gross−Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler−Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures

On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column

A quadrature-based conditional moment closure for mixing-sensitive reactions

A novel algorithm consisting of a quadrature-based semi-analytical solution to the conditional moment closure (CMC) is developed for mixing-sensitive reactions in turbulent flows. When applying the proposed algorithm, the additional grid in mixture-fraction phase space used in CMC codes is eliminated, and at most ten quadrature nodes are needed to model mixing-sensitive turbulent reacting flows. In this work, the mixture-fraction probability density function (PDF) is assumed to be a β-PDF, which is the weight function for the Gauss-Jacobi quadrature rule. The conditional moments of reacting species are determined from unconditional moments that are first order with respect to the species and higher order with respect to mixture fraction. Here, the focus is on the efficient treatment of the molecular-mixing step by using a semi-analytical solution in the form of a Jacobi polynomial expansion. The application of the algorithm is demonstrated considering mixing-sensitive competitive-consecutive and parallel reactions in a statistically homogeneous chemical reactor

Particle-resolved simulation of freely evolving particle suspensions: Flow physics and modeling

The objective of this study is to understand the dynamics of freely evolving particle suspensions over a wide range of particle-to-fluid density ratios. The dynamics of particle suspensions are characterized by the average momentum equation, where the dominant contribution to the average momentum transfer between particles and fluid is the average drag force. In this study, the average drag force is quantified using particle-resolved direct numerical simulation in a canonical problem: a statistically homogeneous suspension where an imposed mean pressure gradient establishes a steady mean slip velocity between the phases. The effects of particle velocity fluctuations, particle clustering, and mobility of particles are studied separately. It is shown that the competing effects of these factors could decrease, increase, or keep constant the drag of freely evolving suspensions in comparison to fixed beds at different flow conditions. It is also shown that the effects of particle clustering and particle velocity fluctuations are not independent. Finally, a correlation for interphase drag force in terms of volume fraction, Reynolds number, and density ratio is proposed. Two different approaches (symbolic regression and predefined functional forms) are used to develop the drag correlation. Since this drag correlation has been inferred from simulations of particle suspensions, it includes the effect of the motion of the particles. This drag correlation can be used in computational fluid dynamics simulations of particle-laden flows that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior

A fully coupled fluid-particle flow solver using quadrature-based moment method with high-order realizable schemes on unstructured grids

Kinetic Equations containing terms for spatial transport, gravity, fluid drag and particle-particle collisions can be used to model dilute gas-particle flows. However, the enormity of independent variables makes direct numerical simulation of these equations almost impossible for practical problems. A viable alternative is to reformulate the problem in terms of moments of the velocity distribution function. A quadrature method of moments (QMOM) was derived by Desjardins et al. [1] for approximating solutions to the kinetic equation for arbitrary Knudsen number. Fox [2, 13] derived a third-order QMOMfor dilute particle flows, including the effect of the fluid drag on the particles. Passalacqua et al. [4] and Garg et al. [3] coupled an incompressible finite-volume solver for the fluid-phase and a third order QMOM solver for particle-phase on Cartesian grids. In the current work a compressible finite-volume fluid solver is coupled with a particle-phase solver based on third-order QMOM on unstructured grids. The fluid and particle-phase are fully coupled by accounting for the volume displacement effects induced by the presence of the particles and the momentum exchange between the phases. The success of QMOM is based on the moment inversion algorithm that allows quadrature weights and abscissas to be computed from the moments of the distribution function. The moment-inversion algorithm does not work if the moments are non-realizable, which might lead to negative weights. Desjardins et al. [1] showed that realizability is guaranteed only with the 1st-order finite-volume scheme that has excessive numerical diffusion. The authors [5, 6] have derived high-order finite-volume schemes that guarantee realizability for QMOM. These high-order realizable schemes are used in this work for the particle-phase solver. Results are presented for a dilute gas-particle flow in a lid-driven cavity with both Stokes and Knudsen numbers equal to 1. For this choice of Knudsen and Stokes numbers, particle trajectory crossing occurs which is captured by QMOM particle-phase solver

A quadrature-based moment method for polydisperse bubbly flows

A computational algorithm for polydisperse bubbly flow is developed by combining quadrature-based moment methods (QBMM) with an existing two-fluid solver for gas–liquid flows. Care is taken to ensure that the two-fluid model equations are hyperbolic by generalizing the kinetic model for the bubble phase proposed by Bieseuvel and Gorissen (1990). The kinetic formulation for the bubble phase includes the full suite of interphase momentum exchange terms for bubbly flow, as well as ad hoc bubble–bubble interaction terms to model the transition from isolated bubbles to regions of pure air at very high bubble-phase volume fractions. A robust numerical algorithm to couple the QBMM approach with a gas–liquid two-fluid solver is proposed. The resulting algorithm is tested to show hyperbolicity, verified against the two-fluid model currently implemented into OpenFOAM, and validated against two sets of experiments on bubbly flows from the literature. In both cases, the computational method shows good agreement with experimental data, and improved accuracy in comparison to a two-fluid model considered for comparison purposes. The robustness of the algorithm is demonstrated on an unstructured mesh with a high superficial gas inlet velocity and source terms for coalescence and breakup. The resulting computational approach is implemented in the open-source CFD code OpenFOAM as part of the OpenQBMM project

Contaminant transport at large Courant numbers using Markov matrices

Volatile organic compounds, particulate matter, airborne infectious disease, and harmful chemical or biological agents are examples of gaseous and particulate contaminants affecting human health in indoor environments. Fast and accurate methods are needed for detection, predictive transport, and contaminant source identification. Markov matrices have shown promise for these applications. However, current (Lagrangian and flux based) Markov methods are limited to small time steps and steady-flow fields. We extend the application of Markov matrices by developing a methodology based on Eulerian approaches. This allows construction of Markov matrices with time steps corresponding to very large Courant numbers. We generalize this framework for steady and transient flow fields with constant and time varying contaminant sources. We illustrate this methodology using three published flow fields. The Markov methods show excellent agreement with conventional PDE methods and are up to 100 times faster than the PDE methods. These methods show promise for developing real-time evacuation and containment strategies, demand response control and estimation of contaminant fields of potential harmful particulate or gaseous contaminants in the indoor environment

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

A moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collisionthe Bhatnagar-Gross-Krook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system with finite-size particles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions

A Delayed Detached Eddy Simulation Model with Low Reynolds Number Correction for Transitional Swirling Flow in a Multi-Inlet Vortex Nanoprecipitation Reactor

The objective of the presented work is to verify a delayed detached eddy simulation (DDES) model for simulating transitional swirling flow in a micro-scale multi-inlet vortex reactor (MIVR). The DDES model is a k-w based turbulence model with a low Reynolds number correction applied to the standard k-w model such that the Reynolds-averaged Navier-Stokes (RANS) component of the DDES model is able to account for low Reynolds number flow. By limiting the dissipation rate in the k-equation, the large-eddy simulation (LES) part of the DDES model behaves similarly to a one-equation sub-grid model. The turbulent Reynolds number is redefined to represent both modeled and resolved turbulence level so that underestimation of the RANS length scale in the LES range can be reduced. Applying the DDES model to simulate both laminar and transitional flow in the micro-scale MIVR produces an accurate prediction of mean velocity and turbulent intensity compared with experimental data. It is demonstrated that the proposed DDES model is capable of simulating transitional flow in the complex geometry of the micro-scale MIVR. These simulation results also help to understand the flow and mixing patterns in the micro-scale MIVR and provide guidances to optimize the reactor for the application of producing functional nanoparticles